That phrase -- "cutting out" -- assumes that the current diet is "normal" ... which it is not. It's a highly artificial construct, which could not have existed even 30 years ago, let alone before the advent of agriculture. IOW, not for almost all of the past 2.5 million years of our existence.
Anyone who knows even schoolboy-level evolutionary biology realises that we are shaped by our environmental conditions, and you can't eat a particular type of diet for hundreds of millions of years without being adjusted to it.
The real question is whether our species can continue to add a higher and higher percentage of foods that we're really not "designed" for to the human diet without virtually killing ourselves off. And I mean that literally.
The current diet has brought about a worldwide epidemic of obesity and diabetes. There are obese five-year olds with plaqued arteries in the States, and there are now children being born already insulin-resistant, owing to the mothers' diets. And obesity and diabetes are only a part of it. Auto-immune diseases are the number three killer in industrialised countries. Anthropologists and medical-missionaries, like Schweitzer, reported not seeing a
single case of cancer among primitive peoples eating their traditional diets. It began to appear as they became westernised. There is a whole cluster of these diseases that were long known by anthropologists as "the diseases of civilisation".
The current diet benefits almost no-one except the supermarkets and the argicultural interests that lie behind the absurd dietary recommendations of the United States Department of Agriculture -- which seem to be taken as a worldwide standard by the ignorant despite the fact that leading scientists have said that they "ignore the science" and "fly in the face of the evidence":
http://www.nutritionjrnl.com/article/PIIS0899900710002893/fulltext